Logical Methods: The Art of Thinking Abstractly and Mathematically

Read about my books: Logical Methods: The Art of Thinking Abstractly and Mathematically and Study book for Logical Methods, both published by The University Press.

Logical Methods

Will be published in English by Springer in February 2021!

Hurray! The book will be published in English by Springer in February 2021! More information is coming!

The purpose of the book is to provide a solid foundation for a study in the sciences, and to introduce and explain the most important and essential concepts within mathematics and science.

Many think mathematics is only about calculations, formulas, numbers, and strange letters. But mathematics is much more than performing calculations, manipulating symbols, and putting numbers into formulas. Mathematics is about discovering patterns, reasoning, finding counterexamples, and arguing logically. Mathematics is a way of thinking. It is an activity that is both extremely creative and challenging.

This book is intended as an introduction to scientific, in particular mathematical, thinking for entry level university or college students. The purpose of the book is to provide a solid foundation for a study in the sciences. It is designed to introduce and explain the most important, essential concepts within mathematics and science. Through 25 short chapters, in an original and visual presentation, the students will learn the basics within set theory, logic, proof methods, combinatorics, graph theory, and much more.

In the book you will find, among other things, answers to:

- What does it mean to abstract and to understand?
- What is a proof and a counterexample?
- What does it mean that something follows logically from a set of premisses?
- How can knowledge and information be represented and calculated with?
- What is the connection between Morse code and Fibonacci numbers?
- Why may it take billions of years to solve Hanoi's Tower?

"Logical methods" is especially appropriate for those who have not studied before, such that the transition to a university or college studies becomes easier. The book also has transfer value to other studies and is a useful and fascinating entry to scientific and logical thinking.

Roger Antonsen received The University Press Textbook Prize in 2013 for this book.

Table of Contents

Chapter 0: The Art of Thinking Abstractly and Mathematically
In this chapter we look at some basic concepts that we will meet over and over again: truths, definitions, assumptions, proofs, languages, axioms, and theorems. We also lay some groundwork and make some conceptual clarifications for the road ahead.
Content Abstraction Reasoning About Truth Assumptions Language Definitions Proofs Problem Solving and Pólya's Heuristics
Chapter 1: Basic Set Theory
In this chapter you will learn the basic concepts and notation used in set theory: what a set is; what operations we perform on sets, such as intersection, union, and set difference; and how sets can be constructed and compared. You will also learn about tuples and Cartesian products.
Content First Steps What Is a Set? Building Sets Operations on Sets Visualizing Sets Comparing Sets Tuples and Products Multisets
Chapter 2: Propositional Logic
In this chapter you will learn about the concept of a proposition, to represent statements in a proper way using propositional formulas, and to understand the connection between propositions (atomic propositions combined by means of logical words) and formulas (propositional variables that are combined using connectives). You will also learn about necessary and sufficient conditions, and some practical abbreviations.
Content What Follows from What? What Is a Proposition? Atomic and Composite Propositions Atomic and Composite Formulas Necessary and Sufficient Conditions Parentheses, Precedence Rules, and Practical Abbreviations
Chapter 3: Semantics for Propositional Logic
In this chapter you will learn semantics for propositional logic. That is, you will learn how to systematically give interpretations to propositional formulas. An assignment of truth values to propositional variables gives rise to a valuation, and this provides truth values for all formulas. You will learn about truth tables, a tool for exploring valuations, about the concept of logical equivalence, and some known logical equivalences.
Content Interpretation of Formulas Valuations and Truth Tables Properties of Implication Logical Equivalence A Study in What Is Equivalent
Chapter 4: Concepts in Propositional Logic
In this chapter you will learn more about semantics for propositional logic, including the concept of logical consequence. A valid argument is one whose conclusion is a logical consequence of the premises. You will also learn about the concepts of satisfiability, falsifiablility, validity, and contradiction, and how these are interrelated.
Content Logical Consequence Valid Arguments Satisfiability and Falsifiability Tautology/Validity and Contradiction Symbols for Truth Values Connections Between Concepts Independence of Formulas Deciding Whether a Formula Is Valid or Satisfiable
Chapter 5: Proofs, Conjectures, and Counterexamples
In this chapter you will learn some methods of proof and become familiar with the relationships between proofs, conjectures, and counterexamples. The goal is to become better at proving statements. Several common methods of proof are discussed: direct proofs, existence proofs, proofs for universal statements, contrapositive proofs, proofs by contradiction, proofs that something is not true, proofs by cases, as well as the difference between constructive and nonconstructive proofs.
Content Proofs Conjectures Thinking from Assumptions Direct Proofs Existence Proofs Proofs by Cases Proofs of Universal Statements Counterexamples Contrapositive Proofs Proofs by Contradiction Constructive Versus Nonconstructive Proofs Proofs of Falsity
Chapter 6: Relations
The aim of this chapter is to understand what relations are and to become familiar with the most important properties of relations, such as reflexivity, symmetry, transitivity, antisymmetry, and irreflexivity. You will also learn about what makes a relation an equivalence relation, a partial order, or a total order.
Content Abstraction over Relations Some Special Relations The Universe of Relations Reflexivity, Symmetry, and Transitivity Antisymmetry and Irreflexivity Orders, Partial and Total Examples
Chapter 7: Functions
In this chapter you will learn about what functions are, as well as the most important concepts related to functions, such as whether they are injective, surjective, or bijective. You will also learn about functions with multiple arguments, how functions can be composed, what operations are, and how functions can be objects.
Content What Is a Function? Injective, Surjective, and Bijective Functions Functions with Multiple Arguments The Universe of Functions Composition of Functions Operations Functions as Objects Partial Functions
Chapter 8: A Little More Set Theory
In this chapter you will learn a little more set theory. You will learn about the universal set, the complement of a set, power sets, and more about Venn diagrams. Finally, we will talk about infinity, cardinality, and the concepts of countability and uncountability.
Content Set Theory Set Complement and the Universal Set Computing with Venn Diagrams Venn Diagrams for Multiple Sets Power Sets Infinity Cardinality Countability Uncountability
Chapter 9: Closures and Inductively Defined Sets
In this chapter you will learn about closures, both of sets and of relations, and how sets can be constructed inductively. We will look at how several sets are defined inductively in this way: sets of numbers, bit strings, propositional formulas, lists, binary trees, and programming languages. In addition, we will become familiar with alphabets and strings, and how formal languages can be defined inductively.
Content Defining Sets Step by Step Closures of Sets Closures of Binary Relations Inductively Defined Sets Sets of Numbers Propositional Formulas Lists and Binary Trees Programming Languages Alphabets, Characters, Strings, and Formal Languages Bit Strings Two Interesting Constructions
Chapter 10: Recursively Defined Functions
In this chapter you will learn to define functions recursively and how such functions are based on inductively defined sets. We will consider recursively defined functions on a series of sets: sets of numbers, bit strings, propositional formulas, lists, binary trees, and formal languages.
Content A Powerful Tool The Triangular Numbers Induction and Recursion Form, Content, and Placeholders Replacing Equals by Equals Recursively Defined Functions Number Sets Bit Strings Propositional Formulas Lists Binary Trees Formal Languages
Chapter 11: Mathematical Induction
In this chapter you will learn about mathematical induction, a powerful and useful method of proof for statements about natural numbers. You will see many examples of proof by induction and learn about how recursively defined functions and proofs by induction are related. You will also learn about the mathematical game Tower of Hanoi and how it can be analyzed using recursively defined functions and proofs by induction.
Content A Mathematical Experiment Mathematical Induction Back to the Experiment A Geometric Proof of the Same Claim What Really Goes On in an Induction Proof? Trominoes Properties of Recursively Defined Functions The Tower of Hanoi More Summing of Numbers Reasoning and Strong Induction
Chapter 12: Structural Induction
In this chapter you will learn about structural induction, a generalization of mathematical induction that works for all inductively defined sets. You will learn to use this method to prove statements about bit strings, propositional formulas, lists, and binary trees.
Content Structural Induction Structural Induction on Bit Strings Structural Induction on Propositional Formulas Structural Induction on Lists Structural Induction on Binary Trees
Chapter 13: First-Order Languages
In this chapter you will learn about the syntax of first-order logic. You will learn about first-order languages; logical and nonlogical symbols; constant, function, and relation symbols; as well as signatures, terms, formulas, and precedence rules.
Content Languages with Greater Expressibility First-Order Languages and Signatures First-Order Terms Prefix, Infix, and Postfix Notation First-Order Formulas Precedence Rules
Chapter 14: Representation of Quantified Statements
In this chapter you will learn about predicates, properties related to free variables, and how first-order languages can be used to represent quantified propositions.
Content Representation of Predicates Syntactic Properties of Free Variables The Art of Expressing Yourself with a First-Order Language Choice of First-Order Language Repeating Patterns in Representations Repetition of First-Order Languages Expressibility and Complexity
Chapter 15: Interpretation in Models
In this chapter you will learn about the semantics of first-order logic: what models are and how they are used to interpret first-order terms and formulas. You will learn about how substitutions are used in the interpretation of quantified formulas, and about the terms valid, satisfiable, contradictory, and falsifiable.
Content Semantics for First-Order Logic Definition of Model Interpretation of Terms Interpretation of Atomic Formulas Substitutions Interpretation of Composite Formulas Satisfiability and Validity of First-Order Formulas First-Order Languages and Equality A Little Repetition
Chapter 16: Reasoning About Models
In this chapter you will learn more about models in first-order logic. You will learn about logical equivalence and consequence, the interaction between the quantifiers and the connectives, modeling, theories, and axiomatizations, and a little bit about prenex normal form.
Content Logical Equivalence and Logical Consequence The Interaction Between Quantifiers and Connectives First-Order Logic and Modeling Theories and Axiomatizations Some Technical Special Cases Prenex Normal Form and More Equivalences Final Comments
Chapter 17: Abstraction with Equivalences and Partitions
In this chapter you will learn about how equivalence relations, equivalence classes, and partitions are related to each other, and how these can be used for abstraction.
Content Abstracting with Equivalence Relations Equivalence Classes Partitions The Connection Between Equivalence Classes and Partitions
Chapter 18: Combinatorics
In this chapter you will learn basic combinatorics. We will go through some basic counting principles, such as the inclusion--exclusion principle and the multiplication principle, and we will define permutations, ordered selections, combinations, and binomial coefficients.
Content The Art of Counting The Inclusion--Exclusion Principle The Multiplication Principle Permutations Ordered Selection Combinations Repetitions and Overcounting
Chapter 19: A Little More Combinatorics
In this chapter you will learn a little more about combinatorics. We will talk more about the binomial coefficients, and go through an example in detail that leads us to Pascal's triangle. Finally, we will look at how enumeration problems can be systematized.
Content Pólya's Example and Pascal's Triangle Binomial Coefficients Systematization of Counting Problems
Chapter 20: A Bit of Abstract Algebra
In this chapter you will learn some concepts from abstract algebra, in particular a bit more about relations and functions. We will talk about inverse relations and functions, a few properties of operations, such as commutativity, associativity, and idempotency, and a few special properties of elements, such as being an identity or inverse. Finally, we will look at what a group is.
Content Abstract Algebra Inverse Relations and Functions Some Properties of Operations Some Elements with Special Properties Groups
Chapter 21: Graph Theory
In this chapter you will learn basic graph theory. We will define graphs, edges, and vertices, and we will introduce terminology to talk about and categorize graphs in a precise way. You will learn a bit about directed graphs, complete graphs, and the complement of graphs. You will also learn two graph-theoretic results that apply to the vertices of a graph. Finally, we will look at what it means for graphs to be isomorphic.
Content Graphs Are Everywhere What Is a Graph? Graphs as Representations Definitions and Concepts About Graphs Properties of Graphs Two Graph-Theoretic Results Isomorphisms
Chapter 22: Walks in Graphs
In this chapter you will learn more about graph theory, especially about different ways to walk around in graphs. We will define walks, paths, trails, circuits, cycles, and trees. Through these concepts you will learn more about basic properties of graphs.
Content The Bridges of Königsberg Paths and Circuits Eulerian Trails and Circuits Hamiltonian Paths and Cycles Final Comments
Chapter 23: Formal Languages and Grammars
In this chapter you will learn more about formal languages. You will learn about the relationship between regular languages and regular expressions, how deterministic finite automata can be used to characterize regular languages, and finally a bit about regular and context-free grammars.
Content Formal Language Theory Operations on Languages Regular Languages Regular Expressions Interpretation of Regular Expressions Deterministic Automata Automata and Regular Languages Nondeterministic Automata Formal Grammars
Chapter 24: Natural Deduction
In this chapter you will learn about a logical calculus called natural deduction. This calculus is a formalization of logical reasoning and can be used to make derivations and proofs of propositional formulas. You will also learn more about the relationship between syntax and semantics via the concepts ofsoundness, completeness, and consistency.
Content Logical Calculi: From Semantics to Syntax Inference Rules of Natural Deduction Closing of Assumptions Derivations and Proofs Negation and RAA The Rules for Disjunction Soundness, Completeness, and Consistency
The Road Ahead
Content Thanks for reading Some book tips The Classics Introductory Books on Mathematical Thinking Introductory Books on Logic Introductory Books on Discrete Mathematics Popular Science, Recreational Mathematics, and Other Books

The University Press Textbook Prize

For the person that is to learn a subject, the subject arises as it is communicated. That is why the textbook is an important book for the learner, and that is precisely why it is difficult to write a textbook. The jury has considered a number of project sketches, and one project excelled. This year's recipient of the Textbook Prize has a doctorate in his field, he is also an award winning science communicator, but he is awarded the prize primarily because of his ability to create the subject in his textbook writing. He communicates with words a subject normally associated with numbers, he shows how mathematics is about understanding the world, and invites the reader to see mathematical activity as a creative and truth-seeking process.

Read more at the home pages of The University Press (in Norwegian).

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Book info

titleLogical Methods: The Art of Thinking Abstractly and Mathematically
authorRoger Antonsen
publisherThe University Press (Universitetsforlaget)
isbn978-82-150-2274-1
year2014
urlThe University Press (Universitetsforlaget)

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Cover and Illustrations

Cover, Marie Låte, 2014
Cover, Marie Låte, 2014

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Illustration from chapter 10 on recursive functions.
Illustration from chapter 10 on recursive functions.
Illustration from chapter 17 on abstraction, partitions, and equivalence classes.
Illustration from chapter 17 on abstraction, partitions, and equivalence classes.